On the extension property of Reifenberg-flat domains

We provide a detailed proof of the fact that any open set whose boundary is sufficiently flat in the sense of Reifenberg is also Jones-flat, and hence it admits an extension operator. We discuss various applications of this property, in particular we obtain L-infinity estimates for the eigenfunctions of the Laplace operator with Neumann boundary conditions. We also compare different ways of measuring the "distance" between two Reifenberg-flat domains. These results are pivotal to the quantitative stability analysis of the spectrum of the Neumann Laplacian performed in [27].